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While the notion of evenness is generally undisputed, its opposite, unevenness, is vague. Conventionally, evenness is understood to decrease, and thus unevenness to increase, as the effective number of types of positive frequency tends towards one while the number of types is held constant. Complete unevenness can thus be approached but never realized for more than one type. In order to arrive at explicit states of minimum evenness for any number of types, equalness is introduced as a complementary concept of evenness. Realization of the new concept turns out to require a change in orientation from unevenness to unequalness. Decreasing evenness in the sense of increasing unequalness entails an increase in inequality among type frequencies. Unequalness can effectively be envisioned as the diversity in the distribution of step‐heights in a frequency distribution ranked in decreasing order (frequency profile). Application of appropriate measures of diversity reveals that maximum unequalness (minimum equalness) is assumed for linearly decreasing frequency profiles (stepladders). A consistent diversity‐based measure of equalness is obtained by normalizing the step‐height diversity with respect to the number of types, with the result that this measure equals one for complete equalness/evenness (uniformity) and zero for complete unequalness (as in stepladders). It turns out that the new equalness measure is highly sensitive to characteristics of frequency profiles that are commonly associated with the evenness notion but are poorly reflected by the conventional evenness measures as a consequence of the realization of their minimum for monomorphism.

While evenness is understood to be maximal if all types (species, genotypes, alleles, etc.) are represented equally (via abundance, biomass, area, etc.), its opposite, maximal unevenness, either remains conceptually in the dark or is conceived as the type distribution that minimizes the applied evenness index. The latter approach, however, frequently leads to conceptual inconsistency due to the fact that the minimizing distribution is not specifiable or is monomorphic. The state of monomorphism, however, is indeterminate in terms of its evenness/unevenness characteristics. Indeed, the semantic indeterminacy also shows up in the observation that monomorphism represents a state of pronounced discontinuity for the established evenness indices. This serious conceptual inconsistency is latent in the widely held idea that evenness is an independent component of diversity. As a consequence, the established evenness indices largely appear as indicators of relative polymorphism rather than as indicators of evenness. In order to arrive at consistent measures of evenness/unevenness, it seems indispensable to determine which states are of maximal unevenness and then to assess the position of a given type distribution between states of maximal evenness and maximal unevenness. Since semantically, unevenness implies inequality among type representations, its maximum is reached if all type representations are equally different. For given number of types, this situation is realized if type representations, when ranked in descending order, show equal differences between adjacent types. We term such distributions “stepladders” as opposed to “plateaus” for uniform distributions. Two approaches to new evenness measures are proposed that reflect different perspectives on the positioning of type distributions between the closest stepladders and the closest plateaus. Their two extremes indicate states of complete evenness and complete unevenness, and the midpoint is postulated to represent the turning point between prevailing evenness and prevailing unevenness. The measures are graphically illustrated by evenness surfaces plotted above frequency simplices for three types, and by transects through evenness surfaces for more types. The approach can be generalized to include variable differences between types (as required in analyses of functional evenness) by simply replacing types with pairs of different types. Pairs, as the new types, can be represented by their abundances, for example, and these can be modified in various ways by the differences between the two types that form the pair. Pair representations thus consist of both the difference between the paired types and their frequency. Omission of pair frequencies leads to conceptual ambiguity. Given this specification of pair representations, their evenness/unevenness can be evaluated using the same indices developed for simple types. Pair evenness then turns out to quantify dispersion evenness.

The metric of functional evenness FEve is an example of how approaches to conceptualizing and measuring functional variability may go astray. This index has several critical conceptual and practical drawbacks: Different values of the FEve index for the same community can be obtained if the species have unequal species abundances; this result is highly likely if most of the traits are categorical. Very minor differences in even one pairwise distance can result in very different values of FEve. FEve uses only a fraction of the information contained in the matrix of species distances. Counterintuitively, this can cause very similar FEve scores for communities with substantially different patterns of species dispersal in trait space. FEve is a valid metric only if all species have exactly the same abundances. However, the meaning of FEve in such an instance is unclear as the purpose of the metric is to measure the variability of abundances in trait space. We recommend not using the FEve metric in studies of functional variability. Given the wide usage of FEve index over the last decade, the validity of the conclusions based on those estimates is in question. Instead, we suggest three alternative metrics that combine variability in species distances in trait space with abundance in various ways. More broadly, we recommend that researchers think about which community properties (e.g., trait distances of a focus species to the nearest neighbor or all other species, variability of pairwise interactions between species) they want to measure and pick from among the appropriate metrics. The index of functional evenness FEve has critical conceptual and practical drawbacks. We recommend not using FEve metric in studies of functional variability. Instead, we suggest three alternative metrics that combine variability in species distances in trait space with abundance in various ways. Different values of the FEve index for the same three communities.

Background Declining resources due to climate change may endanger the persistence of populations by reducing fecundity and thus population fitness via effects on gamete production. The optimal mode of generative reproduction allocates the limited resources to ovule and pollen production in proportions that maximize the number of fertilized ovules in the population. In order to locate this optimum and derive reproduction modes that compensate for declined resources to maintain reproductive success, a model of gamete production, pollen dispersal, and ovule fertilization is developed. Specification of opportunities for compensation is given priority over specification of physiological or evolutionary mechanisms of adaptation. Thus model parameters summarize gametic production resources, resource investment per gamete, resource allocation as proportion of resources invested in ovules, and pollen density as size of the pollen dispersal range and proportion of pollen retained within the range. Retained pollen disperses randomly, and an ovule is fertilized if at least one pollen settles on its surface. The outcome is the expected number of fertilized ovules. Results Maximization of fertilization success is found to require the investment of more gametic production resources in ovules than in pollen, irrespective of the parameter values. Resource decline can be compensated by adjusting the resource allocation if the maximum expected number of fertilized ovules after the decline is not less than the expected number the population experienced before the decline. Compensation is also possible under some conditions by increasing the pollen density, either by raising a low pollen retention or by shrinking the dispersal range. Conclusion Fertilization success in populations affected by resource decline may be maintainable by adjustment of the sexual allocation of gametic production resources or by increasing pollen density. The results have implications for insect pollination, sexual allocation bias, management measures, and metapopulation fragmentation.

(1) Background: Is variation among the communities of a metacommunity higher than within the communities? Community ecologists and population geneticists often characterize the structure of metacommunities by partitioning variation (diversity) into the two following components using measures such as FST or GST and α- and β-diversity. The within-communities component is usually some average of (type, species, genetic) diversities within the communities, and the among-communities component is the additive or multiplicative complement of the overall diversity. Such an among-communities component lacks independent conceptual specification, a matter of long-standing dispute. Only if the two components are independently and commensurably specified can the central question of comparability be answered meaningfully. (2) Methods: A novel approach to overcoming this conceptual weakness identifies two principles of the partitioning of variation among communities (concentration and division) then relates these principles to the common notions of variation (diversity) within and among communities, distinguishes primary indicators to quantify the partitioning principles, transforms the indicators into conceptually independent measures (indices) of variation within and among communities, and by this attains their commensurability and thus comparability. The application of the methods to quantifying the effects of evolutionary mechanisms is outlined. (3) Results: Common approaches are corrected and extended. (a) Analyses of metacommunity/metapopulation structures that rely on apportionment or related indices and take its complement to be differentiation yield incomparable measures of variation within and among communities. (b) The common practice of partitioning the total diversity into additive or multiplicative components produces the inconsistent ranking of the two components. (c) Community concentration and division can result from elementary processes of adaptive differentiation and migration (gene flow) among communities, where the (commensurable) amounts of community concentration and division reflect the relative participation of these processes in metacommunity structuring and translate directly into the measures of diversity within and among communities. (d) The modelling of the contributions of the two partitioning principles to the metacommunity structure is restricted by the marginal distributions of types and community affiliation. (e) The model demonstrates the degree to which adaptational processes at the metacommunity level are mixtures of adaptational events within and among communities.

Diversity in metacommunities is traditionally viewed to consist of the diversity within communities ( α ) that is complemented by the differences between communities ( β ) so as to result in the total diversity ( γ ) of the metacommunity. This perception of the partitioning of diversity, where β is a function of γ and α (usually β = γ / α with all components specified as effective numbers), has several drawbacks, among which are (1)  α is an average that can be taken over communities in many ways, (2) complete differentiation among communities cannot always be uniquely inferred from α and γ , (3) different interpretations of β as effective number of communities (e.g., distinct or monomorphic) are possible, depending on the choice of ideal situations to which the respective effective numbers refer, and (4) associations between types (species, genotypes, etc.) and community affiliations of individuals are not explicitly covered by α and γ . Item (4) deserves special regard when quantifying metacommunity diversity. It is argued that this requires consideration of the joint distribution of type-community combinations together with its diversity (joint diversity) and its constituent components: type and community affiliation. The quantification of both components can be affected by their association as realized in the joint distribution. It is shown that under this perception, the joint diversity can be factorized into a leading and an associated component, where the first characterizes the minimum number of communities required to obtain the observed joint diversity given the observed type distribution, and the second specifies the effective number of types represented in the minimally required number of communities. Multiplication of the two yields the joint diversity. Interchanging the roles of community and type, one arrives at the dual factorization with leading minimum number of types and associated effective number of communities. The two dual factorizations are unambiguously defined for all measures of diversity and can be used, for example, to indicate structural characteristics of metacommunities, such as type differentiation among communities and associated type polymorphism. The information gain of the factorization approach is pointed out in comparison with the classical and more recent modified approaches to partitioning total type diversity into diversity within and between communities. The use of factorization in analyses of latent community subdivision is indicated.

Biological variation is commonly measured at two basic levels: variation within individual communities, and the distribution of variation over communities or within a metacommunity. We develop a classification for the measurement of biological variation on both levels: Within communities into the categories of dispersion and diversity, and within metacommunities into the categories of compositional differentiation and partitioning of variation. There are essentially two approaches to characterizing the distribution of trait variation over communities in that individuals with the same trait state or type tend to occur in the same community (describes differentiation tendencies), and individuals with different types tend to occur in different communities (describes apportionment tendencies). Both approaches can be viewed from the dual perspectives of trait variation distributed over communities (CT perspective) and community membership distributed over trait states (TC perspective). This classification covers most of the relevant descriptors (qualified measures) of biological variation, as is demonstrated with the help of major families of descriptors. Moreover, the classification is shown to open ways to develop new descriptors that meet current needs. Yet the classification also reveals the misclassification of some prominent and widely applied descriptors: Dispersion is often misclassified as diversity, particularly in cases where dispersion descriptor allow for the computation of effective numbers; the descriptor GST of population genetics is commonly misclassified as compositional differentiation and confused with partitioning-oriented differentiation, whereas it actually measures partitioning-oriented apportionment; descriptors of β-diversity are ambiguous about the differentiation effects they are supposed to represent and therefore require conceptual reconsideration.

In a general sense, a metacommunity can be considered as a network of communities, the coherence of which is based on characteristics that are shared by members of different communities, whatever forces were responsible (dispersal, migration, local adaptation, etc.). The purpose is to show that by basing the assessment of coherence on the degree of nestedness of one community within another with respect to the shared characteristics, coherence components can be identified within the network. To assess coherence, a measure of nestedness is developed, and its application to complex (variable) object differences (including multiple traits or characters) is investigated. A community network is then viewed as a graph in which the nodes represent the communities and the edges connecting nodes are weighted by the reverse of the degrees of nestedness between the corresponding communities. Given this framework, it is argued that a minimum requirement for a set of communities to be coherent is the existence of a spanning tree known from graph theory, i.e. a subgraph that connects all nodes through a cycle-free sequence of edges with positive weights. Of all spanning trees, minimum spanning trees (MST, or spanning trees with the minimum sum of edge weights) are most indicative of coherence. By expressing the degree of coherence as one minus the average weight of the edges of an MST, it is uniquely determined which communities form a coherent set at any given level of community distinctness. By this method, community networks can be broken down into coherence components that are separated at a specified distinctness level. This is illustrated in a worked example showing how to apply graph theoretical methods to distinguish coherence components at various threshold levels of object difference (resolution) and community distinctness. These results provide a basis for discussion of coherence gradients and coherence at various levels of distinctness in terms of MST-characteristics. As intuitively expected and analytically confirmed, coherence is a non-decreasing function of the object difference threshold, and the number of coherence components is a non-increasing function of both the object difference and the community distinctness thresholds.

Recently, the notion of diversity, which is directed towards (effective) numbers of types (states of a trait such as species and genotypes), is increasingly used as an umbrella term akin to “variation”, thus including classical metrics of dispersion among others. This is probably due to the growing interest in functional aspects of variation which involve variable differences between types. Though the traditional notion of diversity does not cover these aspects, it shows up in many interpretations. To overcome this ambiguity, the traditional notion of diversity is extended in this paper to include variable differences with emphasis on their general significance as structuring features. For this purpose, structure is conceived to be captured by the representation of types via variable differences and abundances. Structural diversity then results from application of traditional measures of diversity to the relative structural representations of types in addition to their relative abundances. Since diversity as effective number of types alone provides no information about their mutual distinctness and the range covered by them, connections to measures of dispersion are indispensable. This is considered via two approaches that rely on dispersion characteristics and one approach that allows for an assessment of structural diversity for controlled levels of type distinctness. Effects of structure on dispersion and diversity are analyzed. The use of the approaches for discovering rarely considered characteristics of phylogenetic structure is demonstrated.

Environments affect phenotypes through two elementary functions: modifying (by affecting the development of individuals’ phenotypes) and adaptive (by determining the phenotypes’ adaptive significance). Adaptation may be perceived to coordinate the two functions, which may even be performed by the same environmental factor. Organismic transformation of the environment can again affect both functions, where the adaptive functions are commonly addressed via notions of “niche/habitat construction” or “extended phenotype” and modifying function are largely ignored. The multi-causal role of these transformations in evolution and adaptation is hard to model and formalize using standard tools. To arrive at a more comprehensive representation, a systems approach is taken that allows classification and generalization of earlier results and the outlining of new insights. These include the following: ∗ Temporary transformation (restricted to one adaptational episode) is structurally equivalent to adaptation without transformation and therefore provide no new insights. ∗ Prolonged transformation (extending over several episodes) in either adaptive or modifying environments promotes adaptational coordination between the two functions but ultimately prevents persistent adaptedness. ∗ The success of transformations of the adaptive environment that do not affect the modifying environment depends on the diversity in the system states rather than on phenogenetic plasticity. ∗ A substantial difference between transformation of the adaptive and of the modifying environment is that adaptation can be reached within a single episode via transformation of the modifying environment, even if the adaptive environment has no modifying effect. The evolutionary consequences await explicit model analysis. ∗ Migration can be interpreted in terms of environmental transformation of either function, modifying or adaptive, by replacing transformation between environments by migration between them. Established results from migration models can help to reassess existing models of adaptation by environmental transformation and to design new models.